My best short, quick answer is there are two basic vulnerabilities:
one is that the L argument against government is diseconomical; most anything government abandons can and will be done in some fashion by the private sector, but the outcome will often be unsatisfactory. This goes to the idea of public goods in economics (see any textbook on public finance).
two is an argument usefully recapitulated in Bruce Ackerman's Stakeholder Society. There is no meaningful freedom in a context of gross inequality. the whole bit about the poor and the rich having the same right to sleep under bridges. Every runner doesn't begin at the same starting line.
I've debated Cato types quite often but on relatively wonky policy matters. We typically don't get much into first principles. I also work on privatization, but that's also a bit of a diversion, since true libertarians are not much taken with contracting out and all the many other halfway houses between government and totally private (i.e., vouchers, franchising, public enterprises, regulated utilities, etc.). A good general piece on privatization is Paul Starr's report for EPI (http://epinet.org). A good book along the same lines is Robert Kuttner's Everything For Sale.
The L's used to enjoy contrasting the socialist countries lurch away from central planning versus the U.S., but the bloom is clearly off this rose, particularly in Russia. Less spectacular but more relevant to the U.S. is the panoply of problems in public enterprise privatization in the U.K. Another common contrast re: the U.S. and Europe hinges on their higher unemployment rates, about which "Beware the U.S. Model," an EPI book, is useful rebuttal material.
mbs